Consider \(T={T_{\rm SAW}}(w)\)  and the Gibbs distribution \(\mu _T(\cdot \ |\ \Gamma , \sigma )\) , where \(\Gamma \) 
and \(\sigma \in \lbrace \pm 1\rbrace ^{\Gamma }\)  are defined in Section REF .
With respect to the aforementioned Gibbs distribution, let \({{\mathcal {I}}}^{\Gamma ,\sigma }_{T}\)  be the
corresponding pairwise influence matrix.
The following result from [1]} shows a useful relationship between the two
influences matrices \({{\mathcal {I}}}^{\Lambda ,\tau }_{G}\)  and \({{\mathcal {I}}}^{\Gamma ,\sigma }_{T}\) .
